An inverse problem for the three-dimensional multi-connected vibrating membrane with Robin boundary conditions

Zayed, E. M. E.

doi:10.1090/qam/1976367

Author: E. M. E. Zayed
Journal: Quart. Appl. Math. 61 (2003), 233-249
MSC: Primary 35P20; Secondary 35J40, 35R30
DOI: https://doi.org/10.1090/qam/1976367
MathSciNet review: MR1976367
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Abstract: This paper deals with the very interesting problem concerning the influence of the boundary conditions on the distribution of the eigenvalues of the negative Laplacian in ${R^3}$. The trace of the heat semigroup $\theta \left ( t \right ) = \sum \nolimits _{v = 1}^\infty {\exp \left ( - t{\mu _v} \right )}$, where $\left \{ {{\mu _v}} \right \}_{v = 1}^\infty$ are the eigenvalues of the negative Laplacian $- {\nabla ^2} = - {\sum \nolimits _{\beta = 1}^3 {\left ( {\frac {\partial }{{\partial {x^\beta }}}} \right )} ^2}$ in the $\left ( {x^1}, {x^2}, {x^3} \right )$-space, is studied for a general multiply-connected bounded domain $\Omega$ in ${R^3}$ surrounding by simply connected bounded domains ${\Omega _j}$ with smooth bounding surfaces ${S_j}\left ( j = 1,...,n \right )$, where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components $S_i^* \left ( i = 1 + {k_{j - 1}},...,{k_j} \right )$ of the bounding surfaces ${S_j}$ is considered, such that ${S_j} = \cup _{i = 1 + {k_{j - 1}}}^{{k_j}} S_i^*$, where ${k_0} = 0$. Some applications of $\theta \left ( t \right )$ for an ideal gas enclosed in the multiply-connected bounded container $\Omega$ with Robin boundary conditions are given. We show that the asymptotic expansion of $\theta \left ( t \right )$ for short-time $t$ plays an important role in investigating the influence of the finite container $\Omega$ on the thermodynamic quantities of an ideal gas.

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